How to scale a plane to the exact dimensions of a loop cut?
You may do it using a snapping tool. 
First select the edge loop in Edit Mode (Alt+RMB) and snap the cursor to it (Shift+S-->Cursor to Selected). Then select the plane in Object Mode and press Shift+S-->Selection to Cursor [Offset]).


Next press the magnet icon to enable the snapping tool and change its type to Vertex. Scale the plane until it snaps to the loopcut's border. 


To see the loopcut in Object Mode check the Wire and Draw All Edges checkboxes in Display panel of the Object header.


For irregular planes you can scale along the chosen axes (e.g. S,Y).


You may also snap the plane to the corner of the edge loop in Object Mode. Then change the pivot point type to 3D Cursor, snap the cursor to the corner of the plane in Edit Mode with Shift+S-->Cursor to Selected and then scale it until it fits the edgeloop.